Systems of Linear Equations
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalA system is two linear equations holding at once, and its solution is the point where the two lines meet. The algebra is short: substitute one equation into the other, or line them up and eliminate a variable. The points leak in the signs, a substituted negative term flipped positive while combining, and in the reading: a result of no solution or infinitely many misread as the opposite, or the right value reported for the wrong quantity.
These patterns aren't really about whether you can solve a system. They're about whether you carried every sign through the substitution, whether you read what the canceled variables were telling you, and whether you answered the exact quantity the item asked for, not one variable when it wanted the combination.
The work
4 ways in · any order
Lesson
Systems of Linear Equations
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Solve a system by substitution and by elimination, read the number of solutions from the equations and from a graph, then report the exact quantity asked. The lesson works the method and the three patterns that derail it, and it saves the trap for last: the right value for a variable the item never asked about.
Diagnostic
10-item topic check
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Ten items across the three patterns: dropping a sign while substituting, misreading a no-solution or infinitely-many result, and reporting one variable when a combination was asked. A mix of solve-and-report and count-the-solutions questions. Take it cold to surface the ones still catching you, or after the lesson to confirm they are gone.
Grid-in Check
Student-produced response
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About a quarter of SAT math answers are typed, not chosen, with no options to react to. These grid-in items diagnose by the value you enter, then route into the same drills the multiple-choice check feeds.
Targeted Practice
Drill a single pattern
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Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the pattern and you move on.